Logarithmic potentials, quasiconformal flows, and Q-curvature

Abstract

By using quasiconformal flows, we establish that exponentials of logarithmic potentials of measures of small mass are comparable to Jacobians of quasiconformal homeomorphisms of ${\mathbb{R}}^n$, $n\ge 2$. As an application, we obtain the fact that certain complete conformal deformations of an even-dimensional Euclidean space ${\mathbb{R}}^n$ with small total Paneitz or $Q$-curvature are bi-Lipschitz equivalent to standard ${\mathbb{R}}^n$